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# Problem1:The Rockwell hardness of a metal specimen is determined by impressing the surface of the specimen with a hardened point, and then measuring the depth of penetration. The hardness of a certain

Problem1:The Rockwell hardness of a metal specimen is determined by impressing the surface of the specimen with a hardened point, and then measuring the depth of penetration. The hardness of a certain alloy is normally distributed with a mean of 70 units and standard deviation of 3 units.

a)If a specimen is acceptable only if its hardness is between 66 and 74 units, what is the probability that a randomly chosen specimen is acceptable?

b)If the acceptable range is 70 ± c, for what value of c would 95% of all specimens be acceptable?

Problem2:

One thousand independent rolls of a fair die will be made.(a) Find an approximate value of the probability that number 6 will appear between 150 and 200 times inclusively.

(b) If number 6 appears exactly 200 times, estimate the probability that number 5 will appear less than 150 times.

Problem 3. Let X be an exponential random variable with mean 2. Let Y = X5. (a) Write the probability density function of random variable Y .

(b) Find the expected value and the variance for Y .

Problem 4. Let X be a normal random variable with parameters μ and σ (σ > 0), that is the density function of X is given as

1 − (x−μ)2 f(x)= √ e 2σ2 .

σ 2π Show that (a) E[X] = μ and (b) V(X) = σ2.

Problem 5. A code word contains 6 digits: each either 0 or 1: to be valid the word must contain exactly four 1’s and two 0’s. One word is selected at random form the valid code words. Define X1 to be the first (left most digit) and let X2 be the second digit in the word selected.(a) Find the joint probability distribution for X1 and X2 and marginal probabilities of X1 and X2.

(b) Find cov(X1, X2).

Problem 6. The joint density of random variables Xand Y is given by

{

f(x,y) = (a) Find constant C.

Cxy, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1, 0, otherwise.

(b) Find the marginal density of X.

(c) Find the marginal density of Y .

(d) Are X and Y independent? Explain.

(e) Find E[X], E[Y ], E[5X − 3Y ].

(f)FindP(X≤0.5;Y ≥0.75)andP(X≥0.25;0.25≤Y ≤0.5).

Problem 7. Let X1, X2, · · · , Xn be independent random variables, and suppose that Xi has an exponential distribution with mean μi. Find the distribution of Mn = min(X1, X2, · · · , Xn).

Problem 8. Let U ∼ Uniform(0,1). For α > 0 and β > 0, determine the distribution of T =(−βlogU)α.